Finite Group Databasebeta

← C3×#24359 #10642966

Subgroups — C3×#24359 #10642966

Order 768.

1624 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#463C212yesmaximal, complemented
3C2×C3×#447C211yesmaximal, characteristic
4C3×#2436C211yesmaximal, characteristic, complemented
5C3×D4×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2012C212yesmaximal, complemented
8C3×#1893C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1850C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#24359C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16—24no—
16C22×C3×SD16—28no—
17C22×C3×SD16C2212yes—
18C22×C3×SD16C2212yes—
19C22×C3×D8C2211yescharacteristic
20C22×C3×D8C2211yescharacteristic
21C23×C3×C8C2211yescharacteristic
22C2×C3×#124C2212yescomplemented
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C22×C3×C22⋊C4—21no—
26C3×#463—28no—
27C3×#447—24no—
28C2×C3×C4⋊C8C2212yescomplemented
29C2×C3×#103C2212yescomplemented
30C2×C3×#103C2212yescomplemented
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yescomplemented
34C2×C3×C22⋊C8C2211yescharacteristic
35C3×#393—22no—
36C3×#391C2212yescomplemented
37C3×#390C2212yes—
38C3×#386—21no—
39C3×#386—21no—
40C3×#386—22no—
41C3×#380C2211yescharacteristic
42C3×#379—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C23×SD16C2×C311yescharacteristic, complemented
47C2×#463C2×C312yescomplemented
48C2×#447C2×C311yescharacteristic
49#2436C2×C311yescharacteristic, complemented
50D4×C22⋊C4C2×C311yescharacteristic, complemented
51#2055C2×C312yescomplemented
52#2012C2×C312yescomplemented
53#1893C2×C311yescharacteristic, complemented
54#1877C2×C312yescomplemented
55#1850C2×C311yescharacteristic, complemented
56#1787C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—24no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—42no—
63C22×C3×Q8C2312yes—
64C22×C3×D4—24no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—42no—
69C22×C3×D4D414yes—
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4—22no—
75C23×C3×C4C2311yescharacteristic
76C2×C3×SD16—232no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16D418yescomplemented
83C2×C3×D8—22no—
84C2×C3×D8—22no—
85C2×C3×D8—24no—
86C2×C3×D8—42no—
87C22×C3×C8—22no—
88C22×C3×C8—24no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2312yes—
92C3×#124—24no—
93C3×#124—42no—
94C3×#121—22no—
95C3×#121—22no—
96C3×C4×D4—24no—
97C3×C4×D4—42no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—44no—
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C42C2312yes—
111C3×C4⋊C8—24no—
112C3×#103—24no—
113C3×#103—42no—
114C3×#102—22no—
115C3×#102—22no—
116C3×#102—24no—
117C3×C22⋊C8—22no—
118C3×#95—21no—
119C3×#95—22no—
120C3×#95—22no—
121C3×#95D412yes—
122C23×Q8C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×SD16—24no—
125C22×SD16—28no—
126C22×SD16C22×C312yes—
127C22×SD16C22×C312yes—
128C22×D8C22×C311yescharacteristic
129C22×D8C22×C311yescharacteristic
130C23×C8C22×C311yescharacteristic
131C2×#124C22×C312yescomplemented
132C2×#121C22×C311yescharacteristic
133C2×C4×D4C22×C312yes—
134C22×C22⋊C4—21no—
135#463—28no—
136#447—24no—
137C2×C4⋊C8C22×C312yescomplemented
138C2×#103C22×C312yescomplemented
139C2×#103C22×C312yescomplemented
140C2×#102C22×C311yescharacteristic
141C2×#102C22×C311yescharacteristic
142C2×#102C22×C312yescomplemented
143C2×C22⋊C8C22×C311yescharacteristic
144#393—22no—
145#391C22×C312yescomplemented
146#390C22×C312yes—
147#386—21no—
148#386—21no—
149#386—22no—
150#380C22×C311yescharacteristic
151#379—21no—
152C4×C22⋊C4C22×C311yescharacteristic
153#340C22×C312yescomplemented
154#336C22×C312yescomplemented
155C24×C3—21no—
156C24×C3—21no—
157C24×C3—24no—
158C24×C3—21no—
159C24×C3—44no—
160C24×C3—42no—
161C24×C3—42no—
162C24×C3—82no—
163C24×C3C2×D411yescharacteristic
164C2×C3×Q8—22no—
165C2×C3×Q8—28no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—44no—
171C2×C3×Q8—48no—
172C2×C3×Q8—44no—
173C2×C3×Q8—42no—
174C2×C3×Q8C2×D414yes—
175C2×C3×D4—28no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—44no—
187C2×C3×D4—48no—
188C2×C3×D4—44no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4C2×D412yes—
193C2×C3×D4C2×D412yes—
194C2×C3×D4C2×D414yes—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—44no—
201C22×C3×C4—44no—
202C22×C3×C4—44no—
203C22×C3×C4—42no—
204C22×C3×C4—82no—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4#4112yes—
210C22×C3×C4C2411yescharacteristic
211C3×SD16—216no—
212C3×SD16—216no—
213C3×D8—44no—
214C3×D8—44no—
215C2×C3×C8—28no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8C2×D414yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—44no—
230C3×C22⋊C4—44no—
231C3×C22⋊C4—44no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—82no—
236C3×C42—22no—
237C3×C42—22no—
238C25—21no—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—42no—
244C22×Q8C23×C312yes—
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—42no—
250C22×D4C3×D414yes—
251C22×D4C23×C311yescharacteristic
252C22×D4C23×C311yescharacteristic
253C22×D4C23×C311yescharacteristic
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C23×C311yescharacteristic
257C2×SD16—232no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16C3×D418yescomplemented
264C2×D8—22no—
265C2×D8—22no—
266C2×D8—24no—
267C2×D8—42no—
268C22×C8—22no—
269C22×C8—24no—
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C312yes—
273#124—24no—
274#124—42no—
275#121—22no—
276#121—22no—
277C4×D4—24no—
278C4×D4—42no—
279C2×C4⋊C4C23×C312yes—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C3×D411yescharacteristic
290C2×C22⋊C4C3×D411yescharacteristic
291C2×C42C23×C312yes—
292C4⋊C8—24no—
293#103—24no—
294#103—42no—
295#102—22no—
296#102—22no—
297#102—24no—
298C22⋊C8—22no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95C3×D412yes—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—24no—
307C23×C3—22no—
308C23×C3—48no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3—48no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—42no—
322C23×C3—84no—
323C23×C3—88no—
324C23×C3—88no—
325C23×C3#12112yes—
326C23×C3#13512yes—
327C23×C3AGL(1,ℤ/8)12yes—
328C23×C3C22×D411yescharacteristic
329C3×Q8—24no—
330C3×Q8—24no—
331C3×Q8—48no—
332C3×Q8—42no—
333C3×Q8—42no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—48no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—84no—
344C3×D4#12014yes—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—48no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—88no—
366C2×C3×C4#12112yes—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D412yes—
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)12yes—
371C3×C8—24no—
372C3×C8—41no—
373C3×C8—41no—
374C24—21no—
375C24—21no—
376C24—24no—
377C24—21no—
378C24—44no—
379C24—42no—
380C24—42no—
381C24—82no—
382C24C2×C3×D411yescharacteristic
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8C2×C3×D414yes—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×C3×D412yes—
412C2×D4C2×C3×D412yes—
413C2×D4C2×C3×D414yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4C2×C3×D412yes—
425C22×C4C2×C3×D412yes—
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D412yes—
428C22×C4C3×#4112yes—
429C22×C4C24×C311yescharacteristic
430SD16—216no—
431SD16—216no—
432D8—44no—
433D8—44no—
434C2×C8—28no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8C2×C3×D414yes—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—82no—
455C42—22no—
456C42—22no—
457C22×C3—22no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—22no—
461C22×C3—24no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—816no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—81no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3#46314yescomplemented
478C22×C3C2×#12111yescharacteristic
479C22×C3D4212yes—
480C22×C3C2×#13511yescharacteristic
481C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
482C22×C3#57712yes—
483C3×C4—22no—
484C3×C4—44no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—82no—
488C3×C4—81no—
489C3×C4C2×#12011yescharacteristic
490C3×C4#53411yescharacteristic
491C3×C4#54612yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—48no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—48no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—84no—
512C23—88no—
513C23—88no—
514C23C3×#12112yes—
515C23C3×#13512yes—
516C23C3×AGL(1,ℤ/8)12yes—
517C23C22×C3×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4—24no—
527D4—48no—
528D4—41no—
529D4—41no—
530D4—42no—
531D4—42no—
532D4—84no—
533D4C3×#12014yes—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—88no—
555C2×C4C3×#12112yes—
556C2×C4C22×C3×D411yescharacteristic
557C2×C4C22×C3×D412yes—
558C2×C4C2×C3×#4111yescharacteristic
559C2×C4C3×H(5,𝔽2)12yes—
560C8—24no—
561C8—41no—
562C8—41no—
563C2×C3—24no—
564C2×C3—44no—
565C2×C3—82no—
566C2×C3—161no—
567C2×C3#241011yescharacteristic
568C2×C3C2×#46311yescharacteristic
569C2×C3#304211yescharacteristic
570C2×C3#326512yes—
571C2×C3#328212yes—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—24no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—816no—
585C22—82no—
586C22—82no—
587C22—81no—
588C22—82no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22C3×#46314yescomplemented
593C22C2×C3×#12111yescharacteristic
594C22C3×D4212yes—
595C22C2×C3×#13511yescharacteristic
596C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
597C22C3×#57712yes—
598C4—22no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—82no—
603C4—81no—
604C4C2×C3×#12011yescharacteristic
605C4C3×#53411yescharacteristic
606C4C3×#54612yes—
607C3#2435911yescharacteristic, direct product factor, Sylow
608C2—24no—
609C2—44no—
610C2—82no—
611C2—161no—
612C2C3×#241011yescharacteristic
613C2C2×C3×#46311yescharacteristic
614C2C3×#304211yescharacteristic
615C2C3×#326512yes—
616C2C3×#328212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C3×SD16C2semidirect
2. C2×C3×#463C2semidirect
4. C3×#2436C2semidirect
5. C3×D4×C22⋊C4C2semidirect
6. C3×#2055C2semidirect
7. C3×#2012C2semidirect
8. C3×#1893C2semidirect
9. C3×#1877C2semidirect
10. C3×#1850C2semidirect
11. C3×#1787C2semidirect
12. #24359C3direct (N × H)
22. C2×C3×#124C22semidirect
28. C2×C3×C4⋊C8C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#103C22semidirect
33. C2×C3×#102C22semidirect
36. C3×#391C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
46. C23×SD16C2×C3semidirect
47. C2×#463C2×C3semidirect
49. #2436C2×C3semidirect
50. D4×C22⋊C4C2×C3semidirect
51. #2055C2×C3semidirect
52. #2012C2×C3semidirect
53. #1893C2×C3semidirect
54. #1877C2×C3semidirect
55. #1850C2×C3semidirect
56. #1787C2×C3semidirect
82. C2×C3×SD16D4semidirect
131. C2×#124C22×C3semidirect
137. C2×C4⋊C8C22×C3semidirect
138. C2×#103C22×C3semidirect
139. C2×#103C22×C3semidirect
142. C2×#102C22×C3semidirect
145. #391C22×C3semidirect
153. #340C22×C3semidirect
154. #336C22×C3semidirect
263. C2×SD16C3×D4semidirect
477. C22×C3#463semidirect
592. C22C3×#463semidirect
607. C3#24359direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3×SD162
C2×C3×#4632
C2×C3×#4472
C3×#24362
C3×D4×C22⋊C42
C3×#20552
C3×#20122
C3×#18932
C3×#18772
C3×#18502
C3×#17872
#243593